5.8 Questions for the reader

5.8 Questions for the reader#

Exercise 23

Reading a spectrum. A tone is synthesized with additive synthesis using \(f_0 = 100\) Hz, \(K = 3\) harmonics, and amplitudes \(\mathbf{a} = [1, 0, \tfrac{1}{3}]\). Sketch or describe its amplitude spectrum \(|X(f)|\).

  1. At which frequencies are the spikes, and what are their heights?

  2. Which classic waveform shape does this amplitude pattern (odd harmonics only, falling off with harmonic number) most resemble?

Reveal solution

  1. Spikes at \(100\) Hz (height \(1\)) and \(300\) Hz (height \(\tfrac{1}{3}\)), with nothing at \(200\) Hz.

  2. Odd harmonics falling off with harmonic number resemble a square wave.

Exercise 24

Rectangular and polar. Consider the complex number \(z = 1 + j\sqrt{3}\).

  1. Find its magnitude \(r\) and angle \(\theta\), and write it in polar form \(r e^{j\theta}\).

  2. Using the rule that magnitudes multiply and angles add, compute \(z^2\) in polar form and convert back to rectangular form.

Reveal solution

  1. \(r = 2\), \(\theta = \pi/3\), so \(z = 2e^{j\pi/3}\)

  2. \(z^2 = 4e^{j2\pi/3} = -2 + j \cdot 2\sqrt{3}\).

Exercise 25

Phasor projections. A phasor is given by \(2\, e^{j\omega t}\) with frequency \(f = 5\) Hz.

  1. Write expressions for its real and imaginary parts as functions of time.

  2. What is the radius of the circle it traces in the complex plane, and how long does it take to complete one full rotation?

Reveal solution

  1. Real part \(2\cos(10\pi t)\), imaginary part \(2\sin(10\pi t)\)

  2. Radius \(2\); one full rotation every \(0.2\) s.

Exercise 26

Interpreting the transform’s output. Suppose that for some signal, the Fourier transform at a particular frequency \(\omega_0\) evaluates to \(X(\omega_0) = 3 - 4j\).

  1. What is the amplitude \(|X(\omega_0)|\) at that frequency?

  2. What is the phase \(\angle X(\omega_0)\)?

  3. Which of these two numbers would have a larger effect on what the sound is perceived to be, and why?

Reveal solution

  1. \(|X(\omega_0)| = 5\)

  2. \(\angle X(\omega_0) = \arctan(-4/3) \approx -0.93\) rad.

  3. The amplitude matters more perceptually, since hearing is relatively insensitive to phase.

Exercise 27

Inputs and outputs. For each of the following, state its input (domain) and its output (codomain):

  1. A waveform \(x(t)\)

  2. A phasor \(e^{j\omega t}\)

  3. The Fourier transform \(X(\omega)\)

  4. The amplitude spectrum \(|X(\omega)|\)

Reveal solution

  1. Time \(\mathbb{R}\) to amplitude \(\mathbb{R}\).

  2. Time \(\mathbb{R}\) to complex amplitude \(\mathbb{C}\).

  3. Frequency \(\mathbb{R}\) to complex amplitude \(\mathbb{C}\).

  4. Frequency \(\mathbb{R}\) to non-negative amplitude \(\mathbb{R}_{\ge 0}\).

Exercise 28

Which signals have energy at \(\omega\)? Fix a single frequency \(\omega > 0\). For each of the following signals, state whether its Fourier transform has nonzero amplitude \(|F(\omega)|\) at that particular frequency, and briefly justify each answer:

  1. \(3\cos(\omega t - \tfrac{\pi}{4})\)

  2. \(\cos(3\omega t)\)

  3. \(-2\sin(\omega t)\)

  4. \(\cos(-\omega t) + \cos(2\omega t)\)

  5. \(\cos(\omega t) - \cos(\omega t)\)

Reveal solution

  1. Nonzero

  2. Zero

  3. Nonzero

  4. Nonzero

  5. Zero